Polyforms Spirals

The easiest way to define what a polyform spiral is, is with next figure. Here we have geometric art with simple rules. Use two distinct geometric forms and tile the plane totally forming two spirals with each form. Apply colors for better visualization. The process can be applied for certain pairs but not for others. An interesting problem is to find compact spirals for a given pair, that is, spirals with a lot of turns with the fewest pieces possible.

Pentominoes spirals

Pentominoes spirals are plane tiling designs with only two distinct pentominoes. The equal pentominoes remain connected in spiral form. There are some interesting tasks:

1. Get consistent spirals, where the spiral turns number can be extended to the infinity.
2. Get limited spirals, where exists a certain number of maximum of turns. This number could exists as a limit for a pair.
3. Get pentominoes pairs which cannot for spirals of any size.
4. Define a sequence of numbers describing the spiral growing-rate. Lower numbers correspond to compact spirals. Find for a given set of forms which pair has the compactness record.

10-minoes containing more than one pair of equal pentominoes.

An easy way to demonstrate if a pair of pentominoes can form spirals is looking for a plane-tiler 10-minoe containing two copies of each pentomino at the same time. Then, using the tiling, we form a spiral "by hand" as compact as we can and assign one spiral to one pentomino and the other spiral to the other. This way, the pair form an infinite spiral..

By hand, I have found twenty-two of such 10-minoes. Only two of them don't tile the plane (see numbers 3 and 14 below). Next table shows the 66 pairs of two distincts pentominoes. Each cell contains the related 10-minoe(s) for each pair. For instance, 10-minoe number 1 appears in column F, row L, column F row W and column L-row W.

F      I  1 8, 9  L  2 10 9, 12, 13  P  3, 4, 5 9, 10, 11   10, 18  N      13, 14 13 19  T      15 18 18, 20    U      13, 16, 17 13   13    V  1, 6   1 18 18, 21   18    W  7                  X  2, 6, 7 10, 11 14, 17 2, 10 10, 11 14 22 17 6 7  Y  2,5   14 2 5 14         2, 14  Z  1. FLW 7. FXY 2. FPYZ 8. IL 3. FN 9. ILP 4. FN 10. IPNY 5. FNZ 11. INY 6. FWY 12. LP

13. LPTV 14. LTYZ 15. LU 16. LV 17. LVY 18. PNUW 19. NT 20. NU 21. NW 22. UY

The cells marked with color cyan indicate the two pentominoes are in the same plane-tiler 10-minoe and thus a spiral of them exists. There are cases where 10-minoe number 14 appears alone in a cells, pairs LZ, TY and TZ. Since 10-minoe number 14 do not tile the plane, these three cases do not form spirals (yet) according to this simple method. Following figures show some of these spirals. So far, there were found 36 marked cells, so 36 pairs from the 66 form spirals. By the way, I tried to find 15-minoes having three equal pentominoes for more than one type of pentomino but I found nothing.

Spirals FPYZ

The pentominoes F, P, Y and Z can be accommodated in pairs in the same 10-minoe. This 10-minoe can tile the plane in multiple ways. Next diagram shows three spirals for this case. Counting the turns with colored sections we notice each turn require 24 more pieces than the previous one, so 24 indicate the growing-factor for these cases. A single spiral of these demonstrate exists spirals for 6 pairs: FP, FY, FZ, PY, PZ and YZ.

Spirals of Z with F, P or Z

If Z is restricted to be in previous spiral, then the original spiral can be shrinked lowering the growing-factor to 16. Now again, the growing factor seems to be unrelated with tiling variations.

Spiral of I and N, P or Y

Spirals of X and F or Y

More consistent spirals

Spiral PX

This is my best with pair PX. Some parts are repetitive while others don't. Is not simple to demostrate the growing factor will remain as low further.

Other spirals

Spiral LP

LP is the most compact spiral of this page.

Spiral IX

I can't with this pair.

j0r6e m1rele5
may 2003